Risky duration formula for what kind of bond?

Question

In a documentation, there is the following formula for "zero interest rate risky duration" of a bond: $\frac{1-exp(-s \cdot T)}{s}$, where $s$ is spread, $T$ time until maturity.

What type of bond (zero-coupon, floating rate, etc.) is this formula valid for and how is it derived?

Answer 1

Here is my understanding of your question, I might have oversimplified your problem, and made some hypothesis that were not yours.

Assuming we are talking about a bond paying $1$ each day until $T$ or default event if occuring before $T$.

Let's write the risky coupon bond payment in a continous time manner:

$$\int_0^T \mathbb{1}_{\tau>t} dt$$

with zero interest rate, NPV is given by (assuming flat intensity)

$$NPV = \int_{0}^{T} \mathbb{E}[\mathbb{1}_{\tau>t}]dt = \int_{0}^{T} e^{-st}dt=\frac{1-e^{-sT}}{s}$$

We focus on risky duration as derivative of the NPV with respect to spread.

Here, we are in a continuous time framework where we assumed no recovery in NPV, so intensity and spread are the same

$$\text{risky duration}=\frac{dNPV}{ds}=\frac{(1+sT)e^{-sT}-1}{s^2}=-\frac{T^2}{2}+O_{s\to 0}(sT^3)]$$